The replicator dynamics for age structured populations

1 Abstract In this paper is presented the new modelling framework combining the repli-cator dynamics (which is the standard model of the frequency dependent selection) with the Leslie Matrix model of the age-structured population. Firstly the continuous version of the discrete Leslie Matrix model is derived. It is shown that Euler–Lotka equation is satisfied when new model reaches the steady state (ie. stable related frequencies between the age classes). Due to the long expected lifespan of an individual in comparison with the ecological timescale, the real life model should contain a large number of equations. This problem is solved by the introduction of the large age classes concept. The underlying assumption is that within a single large age class the individuals do not differ in the demographic parameters (fertility and mortality). Then according to this result, a more complex model containing different individual strategies is presented. The methodology of the multipopulation games is used for derivation of two, mutually equivalent systems of equations. First contains equations describing the evolution of the strategy frequencies in the whole population completed by subsystems of equations describing the evolution of the age structure for each strategy. Second system contains equations describing the changes of general populations age structure, completed with subsystems of equations describing the selection of the strategies within each age class.